US8248286B2 - Split wiper digital potentiometer and method - Google Patents
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- US8248286B2 US8248286B2 US12/846,668 US84666810A US8248286B2 US 8248286 B2 US8248286 B2 US 8248286B2 US 84666810 A US84666810 A US 84666810A US 8248286 B2 US8248286 B2 US 8248286B2
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M1/00—Analogue/digital conversion; Digital/analogue conversion
- H03M1/66—Digital/analogue converters
- H03M1/68—Digital/analogue converters with conversions of different sensitivity, i.e. one conversion relating to the more significant digital bits and another conversion to the less significant bits
- H03M1/682—Digital/analogue converters with conversions of different sensitivity, i.e. one conversion relating to the more significant digital bits and another conversion to the less significant bits both converters being of the unary decoded type
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M1/00—Analogue/digital conversion; Digital/analogue conversion
- H03M1/66—Digital/analogue converters
- H03M1/74—Simultaneous conversion
- H03M1/76—Simultaneous conversion using switching tree
- H03M1/765—Simultaneous conversion using switching tree using a single level of switches which are controlled by unary decoded digital signals
Definitions
- the present invention relates, in general, to electronics and, more particularly, to potentiometers and methods for manufacturing potentiometers.
- Potentiometers are electronic circuits that provide a variable impedance between two nodes.
- Digital potentiometers are electronic circuits that provide a variable impedance in accordance with a digital signal.
- a digital potentiometer has a fixed value impedance connected between two reference terminals. This impedance is provided by a string of impedance devices that can be selectively connected to a third terminal, called a wiper terminal, through electronic switches that are controlled by digital signals.
- Digital potentiometers can be used in digital-to-analog converters and as replacements for mechanical potentiometers and rheostats.
- the fineness of adjustment, resolution, or “granularity” of a digital potentiometer is typically determined by the number of digital bits used for the selection of the desired wiper position. For instance, an eight bit wiper address allows for 2 8 , i.e., 256, different wiper impedance selections or wiper positions.
- a disadvantage of finer adjustment granularity is a rapid increase in the number of components such as resistors, switches, decoding gates, or other logic devices required for implementation. The increase in the number of components typically results in larger and more expensive devices.
- FIG. 1 is a circuit schematic of a potentiometer in accordance with an embodiment of the present invention
- FIGS. 2A-2B are circuit schematics of a portion of the potentiometer of FIG. 1 ;
- FIGS. 3A-3F are circuit schematics of a portion of a potentiometer having 256 taps in accordance with an embodiment of the present invention
- FIGS. 4A-4B are charts of switch positions of the potentiometer of FIGS. 3A-3F ;
- FIGS. 5A-5B are circuit schematics of a portion of the potentiometer of FIG. 1 in accordance with another embodiment of the present invention.
- the present description includes, among other features, a potentiometer and a method for adjusting an impedance.
- the potentiometer may be a programmable multistage digital potentiometer which preferably has 2 n taps and at least three terminals: a higher terminal, a lower terminal, and a wiper terminal.
- the terms higher and lower for describing the terminals are not limiting terms with regards to the position or function of the terminals.
- the higher terminal is coupled to the lower terminal through a string of impedance elements.
- the potentiometer is a programmable multistage digital potentiometer has 2 n taps, there are 2 n ⁇ 1 impedance elements, where n represents the number of bits used for identifying the position of the wiper terminal.
- This signal may be referred to as the wiper address (WA[n]:WA[0]).
- the impedance element is a resistor and there are (2 n ⁇ 1) impedance elements between the upper and lower terminals
- the nominal resistance value R nom between the upper and lower terminals is (2 n ⁇ 1) times R 0 , where R 0 is a resistance value of a single impedance element.
- the 2 n taps can be connected one at time to the wiper terminal.
- the impedance element may be a resistor, a capacitor, an inductor, or combinations thereof.
- the reactive part (X) is zero whereas for a capacitor and an inductor the resistive part is zero such that the impedance of a capacitor or an inductor is purely imaginary.
- impedance is a term that encompasses both the resistance of resistors and the reactance of capacitors and inductors.
- the impedance elements are resistors
- changing the impedance configuration when the resistive impedance such as a resistor is replaced by an impedance having a reactive part such as a capacitor For instance, changing a string of resistors that are in series may require using a string of capacitors that are in parallel.
- n n MSB +n MED +n LSB EQT. 1
- n MSB is a design parameter for a stage that represents the most significant bits, i.e., the MSB stage;
- n MED is a design parameter for a stage that represents the medium range bits, i.e., the medium range stages;
- n LSB is a design parameter for a stage that represents the least significant bits, i.e., the LSB range stages.
- n MSB is selected to have a minimum value, i.e., one.
- the whole range of impedance values can be divided into an upper range and a lower range.
- n 1+[( n ⁇ 1)/2 ]+[n/ 2] EQT. 4
- a programmable digital potentiometer having 2 n taps has 2 n ⁇ 1 impedance elements.
- the most significant bit (n MSB ) of the n-bit word is selected to divide the range of the wiper address into a higher range and a lower range, i.e., n MSB is selected to be 1.
- n MSB 1 EQT. 5
- n MED [( n ⁇ 1)/2] EQT. 6
- n LSB [n/ 2] EQT. 7
- the number of bits of the wiper address represented by n LSB is 4, i.e., n LSB is represented by 4 of the 8 wiper bits.
- n LSB the number of bits of the wiper address represented by n LSB is 3, i.e., n LSB is represented by 3 of the 7 wiper bits.
- the number of bits of the wiper address represented by n MSB is 1, i.e., 1 of the 7 wiper bits.
- the programmable multistage digital potentiometer can be implemented having a Most Significant Bit (MSB) impedance or low resolution stage, a higher impedance stage coupled between the MSB stage and the higher terminal, and a lower impedance stage coupled between the MSB stage and the lower terminal.
- MSB Most Significant Bit
- a wiper terminal is coupled to the higher and lower impedance stages.
- the higher impedance stage is comprised of a higher medium range impedance stage (also referred to as a medium resolution stage) and a higher Least Significant Bit (LSB) impedance stage (also referred to as a high resolution stage), where the higher medium range impedance stage provides higher medium range impedance values that are selected by the medium range bits n MED and the higher LSB impedance stage provides higher resolution impedance values for the higher impedance range selected by the LSB bits n LSB .
- a higher medium range impedance stage also referred to as a medium resolution stage
- LSB impedance stage also referred to as a high resolution stage
- the lower stage is comprised of a lower medium range impedance stage (also referred to as a medium resolution stage) and a lower LSB impedance stage (also referred to as a high resolution stage), where the lower medium range impedance stage provides lower medium range impedance values that are selected by the medium range bits n MED and the lower LSB impedance stage provides higher resolution impedance values for the lower impedance range selected by the LSB bits n LSB .
- the higher and lower medium impedance stages are symmetrically coupled about the MSB stage and the higher and lower LSB impedance stages are symmetrically coupled about the MSB stage.
- FIG. 1 is a schematic diagram of a programmable multistage digital potentiometer 10 in accordance with an embodiment of the present invention. What is shown in FIG. 1 is a control/decoder block 12 coupled to a variable impedance network 14 . More particularly, control/decoder block 12 includes an interface circuit 16 coupled for receiving a binary input signal at input terminal 18 from an external source.
- the external source may be a device external to an integrated circuit containing digital potentiometer 10 or another circuit within the same integrated circuit as digital potentiometer 10 .
- the input signal at terminal 18 may include an n-bit wiper address and/or other data and/or instructions for digital potentiometer 10 .
- Interface circuit 16 may be a serial interface or a parallel interface.
- Control/decoder block 12 further includes a control logic circuit 20 , a wiper address register 22 , a non-volatile register 24 , and a decoder 26 .
- Control logic circuit 20 is coupled for transmitting and receiving signals to and from interface circuit 16 and coupled for transmitting control signals to wiper address register 22 , non-volatile register 24 , and decoder 26 .
- wiper address register 22 is coupled for transmitting and receiving data signals to and from interface circuit 16 and non-volatile register 24 and transmitting an address signal to decoder 26 .
- Decoder 26 transmits switch control signals to variable impedance network 14 .
- Non-volatile register 24 may be an Electrically Erasable Programmable Read Only Memory (EEPROM), flash memory, or the like. Alternatively, non-volatile register 24 may be omitted from programmable multistage digital potentiometer 10 . Although register 24 has been described as a non-volatile register, this is not a limitation of the present invention. Register 24 may be a volatile device. By way of example, a non-volatile register 24 stores an initial wiper position value that may be provided to switch logic circuit 20 at system start up for implementation by control logic circuit 20 .
- EEPROM Electrically Erasable Programmable Read Only Memory
- the initial wiper position may be a last wiper position of digital potentiometer 10 before an immediately prior turn-off of digital potentiometer 10 , or may be a universal start-up value, e.g., a lowest wiper position, a highest wiper position, or a middle wiper position.
- control logic circuit 20 may be programmed to implement a default wiper position (e.g., a wiper position in the middle of the wiper address range) upon power-on, and then may either increment or decrement the wiper position from the default wiper position, or may jump to a distal (i.e., non-incremental) wiper position from the default position, based on a subsequent input n-bit wiper address.
- a default wiper position e.g., a wiper position in the middle of the wiper address range
- Variable impedance network 14 has a lower reference voltage terminal 30 and a higher reference voltage terminal 32 and is comprised of a string of impedance stages 40 , 42 , and 44 where each impedance stage includes one or more impedance elements.
- Impedance stage 42 has a terminal 42 A connected to a terminal 40 B of impedance stage 40 and a terminal 42 B connected to a terminal 44 A of impedance stage 44 .
- Impedance stage 40 has a terminal that serves as lower reference voltage terminal 30 and impedance stage 44 has a terminal that serves as upper reference voltage terminal 32 .
- a common wiper terminal 34 is connected to impedance stages 40 and 44 through wiper switches 46 and 48 , respectively.
- Impedance stages 40 and 44 operate in response to different bits of an n-bit wiper address, i.e., bits n MED and n LSB as described above and impedance stage 42 has a substantially constant impedance value and its relative position with respect to the common wiper terminal is controlled by a Most Significant (MSB) of the n-bit wiper address.
- Wiper switches 46 and 48 operate in response to the Most Significant Bit of the n-bit wiper address n MSB and are referred to as MSB switches.
- Impedance stage 40 is comprised of two sub-stages 50 and 52 , where sub-stage 50 may be referred to as a high resolution sub-stage and sub-stage 52 may be referred to as a medium resolution sub-stage.
- impedance stage 44 is comprised of two sub-stages 60 and 62 , where sub-stage 60 may be referred to as a high resolution sub-stage and sub-stage 62 may be referred to as a medium resolution sub-stage.
- Impedance stage 40 may be referred to as a lower voltage stage and impedance stage 44 may be referred to as a higher voltage stage because they are connected respectively to lower reference voltage terminal 30 and higher reference voltage terminal 32 .
- high resolution stage 50 operates in response to one or more of the least significant bits of the n-bit wiper address and medium resolution stage 52 operates in response to one or more bits of the n-bit wiper address between the one or more least significant bits and the most significant bit of the n-bit wiper address. Accordingly, high resolution stage 50 may be referred to as a lower voltage Least Significant Bit (LSB) stage and medium resolution stage 52 may be referred to as a lower voltage medium range stage.
- LSB Least Significant Bit
- high resolution stage 60 operates in response to one or more of the least significant bits of the n-bit wiper address and medium resolution stage 62 operates in response to one or more bits of the n-bit wiper address that are between the one or more least significant bits and the most significant bit of the n-bit wiper address. Accordingly, high resolution stage 60 may be referred to as a higher voltage LSB stage and medium resolution stage 62 may be referred to as a higher voltage medium range stage.
- variable impedance network 14 may be referred to as being comprised of three stages, where MSB stage 42 comprises one stage, lower voltage LSB stage 50 and higher voltage LSB stage 60 form another stage, and lower voltage medium range stage 52 and higher voltage medium range stage 62 form another stage.
- FIGS. 2A and 2B a circuit schematic of variable impedance network 14 is shown. It should be understood that FIG. 2A shows a portion of variable impedance network 14 and FIG. 2B shows another portion of variable impedance network 14 . Accordingly, FIGS. 2A and 2B together show an embodiment of variable impedance network 14 . FIGS. 1 , 2 A, and 2 B are described together.
- variable impedance network 14 includes lower voltage impedance stage 40 , MSB stage 42 , and higher voltage impedance stage 44 , where lower voltage impedance stage 40 is comprised of lower voltage impedance LSB structure or stage 50 and lower voltage medium range impedance structure or stage 52 and impedance stage 44 is comprised of higher voltage LSB impedance structure or stage 60 and higher voltage medium range impedance structure or stage 62 .
- lower voltage medium range impedance stage 52 has a lower terminal 52 A, an upper terminal 52 B, and includes a dummy impedance structure Z LD connected to a string of impedance elements Z L ( 1 ), . . . , Z L (k), . . . , Z L ( 2 nMED ⁇ 1 ) and corresponding switches S L ( 1 ), . . . , S L (k), . . . , S L ( 2 nMED ⁇ 1 ).
- k is an integer in the range 2 to 2 nMED ⁇ 2 and is used to denote one or more impedance elements Z L (k) connected in series between impedance elements Z L ( 1 ) and Z L ( 2 nMED ⁇ 1 ) and one or more switches S L (k) that are coupled to corresponding impedances Z L (k).
- Dummy impedance structure Z LD has terminals Z LD1 and Z LD2 ; impedance element Z L ( 1 ) has terminals Z L1 ( 1 ) and Z L2 ( 1 ); impedance element Z L (k) has terminals Z L1 (k) and Z L2 (k); impedance element Z L ( 2 nMED ⁇ 1 ) has terminals Z L1 ( 2 nMED ⁇ 1 ) and Z L2 ( 2 nMED ⁇ 1 ); switch S L ( 1 ) has current conducting terminals S L1 ( 1 ) and S L2 ( 1 ) and a control terminal S L3 ( 1 ), switch S L (k) has terminals S L1 (k) and S L2 (k) and a control terminal S L3 (k); and switch S L ( 2 nMED ⁇ 1 ) has current conducting terminals S L1 ( 2 nMED ⁇ 1 ) and S L2 ( 2 nMED ⁇ 1 ) and a control terminal S L3 ( 2 nMED
- Terminal Z LD1 of dummy impedance structure Z LD is connected to lower reference voltage terminal 30 and terminal Z LD2 is connected to terminal Z L1 ( 1 ) of impedance element Z L ( 1 ).
- Terminal Z L2 ( 1 ) is coupled to terminal Z L1 (k)
- terminal Z L2 (k) is coupled to terminal Z L1 ( 2 nMED ⁇ 1 )
- terminal Z L2 ( 2 nMED ⁇ 1 ) is the upper end of the impedance string connected to node 70 .
- Z L (k) and S L (k) are short hand notations to indicate the presence of one or more impedance elements and one or more switches.
- S L1 ( 2 nMED ⁇ 1 ) of switches S L ( 1 ), . . . , S L (k), . . . , S L ( 2 nMED ⁇ 1 ), respectively, are commonly connected to terminal 30 .
- Current conducting terminal S L2 ( 1 ) of switch S L ( 1 ) is connected to terminal Z L2 ( 1 )
- current conducting terminal S L2 (k) is coupled to terminal Z L2 (k)
- current conducting terminal S L2 ( 2 nMED ⁇ 1 ) is connected to node 70 .
- the number of impedance elements Z L of impedance stage 52 can range from 1 to 2 nMED ⁇ 1.
- impedance structure 52 can range from 2 to 2 nMED ⁇ 2.
- lower voltage medium range impedance stage 40 may be comprised of a dummy impedance structure Z LD and a single impedance element Z L ( 1 ).
- Dummy impedance structure Z LD is a circuit element connected in series with the chain or string of impedance elements Z L ( 1 ) to Z L ( 2 nMED ⁇ 1 ) that provides a constant equivalent impedance between the current conducting terminals S L1 (m) and S L2 (m) of any closed switch, i.e., a switch that is on, if all the other switches S L (j) under it are also closed or shorted. It should be noted that “j” is greater than or equal to 1 and less than or equal to “m.”
- the dummy structure is a resistor connected in parallel with a switch that is permanently on or closed.
- the resistance value of the dummy impedance depends on the value of the equivalent resistance of medium range impedance stage 52 and on the switch series on-resistance, R SH .
- the value of the dummy structure impedance is independent of the number of switches and whether they are shorted or not.
- nMED ⁇ 1 the number of shorted switches in medium range impedance stage 52 and may range from zero to 2 nMED ⁇ 1;
- R MED is a medium resistor value
- R SHORT is an impedance having a constant value representing the equivalent impedance between the current conducting terminals S L1 (m) and S L2 (m) of switch S L (m) for any “m” in the range 1 to 2 nMED ⁇ 1 and where R SHORT is in series with resistor R MED .
- resistor R SHORT ( R MED +( R MED *( R MED +4 *R SH )) 1/2 )/2 EQT. 9
- R SH is the switch series on-resistance.
- a switch S L ( 1 ) having current conducting terminals S L1 ( 1 ) and S L2 ( 1 ) and a control terminal S L3 ( 1 ) is connected to impedance element Z L ( 1 ). More particularly, terminal S L1 ( 1 ) is connected to terminal Z LD1 of dummy impedance element Z LD , terminal S L2 ( 1 ) is connected to terminal Z L2 ( 1 ) of impedance element Z L ( 1 ), and control terminal S L3 ( 1 ) of switch S L ( 1 ) is connected to terminal D L ( 1 ) of decoder 26 .
- a switch S L (k) having current conducting terminals S L1 (k) and S L2 (k) is connected to impedance element Z L (k).
- Terminal S L1 (k) is connected to lower reference voltage terminal 30 and terminal S L2 (k) is connected to terminal Z L2 (k) of impedance element Z L (k).
- Terminal S L1 ( 2 nMED ⁇ 1 ) is connected to lower reference voltage terminal 30 and terminal S L2 ( 2 nMED ⁇ 1 ) is connected to terminal Z L2 ( 2 nMED ⁇ 1 ) of impedance element Z L ( 2 nMED ⁇ 1 ).
- Control terminal S L3 ( 2 nMED ⁇ 1 ) is connected to terminal D L ( 2 nMED ⁇ 1 ) of decoder 26 .
- Connecting current conducting terminal S L2 ( 2 nMED ⁇ 1 ) to terminal Z L2 ( 2 nMED ⁇ 1 ) of impedance element Z L ( 2 nMED ⁇ 1 ) forms a node 70 .
- Lower voltage LSB stage 50 is comprised of one or more impedance elements Z WL ( 1 ), . . . , Z WL (k), . . . , Z WL ( 2 nLSB ⁇ 1 ) and a plurality of switches S WL ( 1 ), S WL ( 2 ), . . . , S WL (k), S WL (k+ 1 ), . . . , S WL ( 2 nLSB ⁇ 1 ), S WL ( 2 nLSB ).
- Impedance element Z WL ( 1 ) has current conducting terminals Z WL1 ( 1 ) and Z WL2 ( 1 )
- impedance element Z WL (k) has current conducting terminals Z WL1 (k) and Z WL2 (k)
- impedance element Z WL ( 2 nLSB ⁇ 1 ) has current conducting terminals Z WL1 ( 2 nLSB ⁇ 1 ) and Z WL2 ( 2 nLSB ⁇ 1 ).
- Switch S WL ( 1 ) has current conducting terminals S WL1 ( 1 ) and S WL2 ( 1 ) and a control terminal S WL3 ( 1 )
- switch S WL ( 2 ) has current conducting terminals S WL1 ( 2 ) and S WL2 ( 2 ) and a control terminal S WL3 ( 2 )
- switch S WL (k) has current conducting terminals S WL1 (k) and S WL2 (k) and a control terminal S WL3 (k)
- switch S WL (k+ 1 ) has current conducting terminals S WL1 (k+ 1 ) and S WL2 (k+ 1 ) and a control terminal S WL3 (k+ 1 )
- switch S WL ( 2 nLSB ⁇ 1 ) has current conducting terminals S WL1 ( 2 nLSB ⁇ 1 ) and S WL2 ( 2 nLSB ⁇ 1 ) and a control terminal S WL3 ( 2 nLSB ⁇ 1
- Current conducting terminal Z WL1 ( 1 ) is connected to node 70 and current conducting terminal Z WL2 ( 1 ) is connected to a current conducting terminal Z WL1 (k).
- Current conducting terminal Z WL1 (k) is coupled to current conducting terminal Z WL2 ( 1 )
- current conducting terminal Z WL2 (k) is coupled to current conducting terminal Z WL1 ( 2 nLSB ⁇ 1 )
- current conducting terminal Z WL2 ( 2 nLSB ⁇ 1 ) which serves as terminal 40 B of lower voltage impedance stage 40 , is coupled to terminal 42 A of MSB stage 42 .
- the number of impedance elements Z WL of impedance stage 50 can range from 1 to 2 nLSB ⁇ 1.
- the index or variable “k” for impedance structure 50 can range from 2 to 2 nLSB ⁇ 2.
- Current conducting terminals S WL1 ( 1 ) and S WL1 ( 2 ) are connected to terminals Z WL1 ( 1 ) and Z WL2 ( 1 ) of impedance element Z WL ( 1 ), respectively.
- Current conducting terminals S WL1 (k) and S WL1 (k+ 1 ) are connected to terminals Z WL1 (k) and Z WL2 (k) of impedance element Z WL (k), respectively.
- S WL2 (k), S WL2 (k+ 1 ), . . . , S WL2 ( 2 nLSB ⁇ 1 ), S WL2 ( 2 nLSB ⁇ 1 ) are commonly connected together to form a node 71 .
- Control terminals S WL3 ( 1 ), S WL3 ( 2 ), . . . , S WL3 (k), S WL3 (k+ 1 ), . . . , S WL3 ( 2 LSB ⁇ 1 ), S WL3 ( 2 nLSB ) are connected to terminals D WL ( 1 ), D WL ( 2 ), . . . , D WL (k), D WL (k+ 1 ), . . . , D WL ( 2 nLSB ⁇ 1 ), D WL 2 nLSB of decoder 26 , respectively.
- lower voltage LSB stage 50 may be comprised of a single impedance element and two switches such as, for example, impedance element Z WL ( 1 ) and switches S WL ( 1 ) and S WL ( 2 ).
- MSB stage 42 is comprised of a bulk impedance having ends that form terminals 42 A and 42 B.
- MSB stage 42 is comprised of a bulk resistance of the semiconductor material from which programmable multistage digital potentiometer is fabricated.
- a switch is absent from MSB stage 42 , thus it is referred to as a non-shunted bulk impedance.
- Higher voltage LSB stage 60 is comprised of one or more impedance elements Z WH ( 1 ), . . . , Z WH (k), . . . , Z WH ( 2 nLSB ⁇ 1 ) and a plurality of switches S WH ( 1 ), S WH ( 2 ), . . . , S WH (k), S WH (k+ 1 ), . . . , S WH ( 2 nLSB ⁇ 1 ), S WH ( 2 nLSB ).
- Impedance element Z WH ( 1 ) has current conducting terminals Z WH1 ( 1 ) and Z WH2 ( 1 )
- impedance element Z WH (k) has current conducting terminals Z WH1 (k) and Z WH2 (k)
- impedance element Z WH1 ( 2 nLSB ⁇ 1 ) has current conducting terminals Z WH1 ( 2 nLSB ⁇ 1 ) and Z WH2 ( 2 nLSB ⁇ 1 ).
- Switch S WH ( 1 ) has current conducting terminals S WH1 ( 1 ) and S WH2 ( 1 ) and a control terminal S WH3 ( 1 );
- switch S WH ( 2 ) has current conducting terminals S WH1 ( 2 ) and S WH2 ( 2 ) and a control terminal S WH3 ( 2 );
- switch S WH (k) has current conducting terminals S WH1 (k) and S WH2 (k) and a control terminal S WH3 (k);
- switch S WH (k+ 1 ) has current conducting terminals S WH1 (k+ 1 ) and S WH2 (k+ 1 ) and a control terminal S WH3 (k+ 1 );
- switch S WH ( 2 nLSB ⁇ 1 ) has current conducting terminals S WH1 ( 2 nLSB ⁇ 1 ) and S WH2 ( 2 nLSB ⁇ 1 ) and a control terminal S WH3 ( 2 nLSB ⁇ 1
- Current conducting terminal Z WH1 ( 1 ) is connected to terminal 44 A (shown in FIG. 1 ) and current conducting terminal Z WH2 ( 1 ) is connected to current conducting terminal Z WH1 (k). Terminals 44 A and 42 B are electrically connected together.
- Current conducting terminal Z WH1 (k) is connected to current conducting terminal Z WH2 ( 1 ) and current conducting terminal Z WH2 (k) is connected to current conducting terminal Z WH1 ( 2 nLSB ⁇ 1 ).
- the number of impedance elements Z WH of impedance stage 62 can range from 1 to 2 nLSB ⁇ 1.
- impedance structure 62 can range from 2 to 2 nLSB ⁇ 2.
- Current conducting terminals S WH1 ( 1 ) and S WH1 ( 2 ) are connected to terminals Z WH1 ( 1 ) and Z WH2 ( 1 ) of impedance element Z WH ( 1 ), respectively.
- Current conducting terminals S WH1 (k) and S WH1 (k+ 1 ) are connected to terminals Z WH1 (k) and Z WH2 (k) of impedance element Z WH (k), respectively.
- S WH2 ( 2 nLSB ⁇ 1 ), S WH2 ( 2 nLSB ) are commonly connected together to form a node 73 .
- Control terminals S WH3 ( 1 ), S WH3 ( 2 ), . . . , S WH3 (k), S WH3 (k+ 1 ), . . . , S WH3 ( 2 nLSB ⁇ 1 ), S WH3 ( 2 nLSB ) are connected to terminals D WH ( 1 ), D WH ( 2 , . . . , D WH (k), D WH (k+ 1 ), . . . , D WH ( 2 nLSB ⁇ 1 ), D WH 2 nLSB of decoder 26 , respectively.
- higher voltage LSB stage 60 may be comprised of a single impedance element and two switches such as, for example, impedance element Z WH ( 1 ) and switches S WH ( 1 ) and S WH ( 2 ).
- higher voltage medium range impedance stage 62 has a lower terminal 62 A, an upper terminal 62 B, and includes a dummy impedance structure Z HD connected to a string of impedance elements Z H ( 1 ), Z H (k), . . . , Z H ( 2 nMED ⁇ 1 ) and corresponding switches S H ( 1 ), . . . , S H (k), . . . , S H ( 2 nMED ⁇ 1 ), where “k” is an integer in the range 2 to 2 nMED ⁇ 2.
- Dummy impedance structure Z HD has terminals Z HD1 and Z HD2 ; impedance element Z H ( 1 ) has terminals Z H1 ( 1 ) and Z H2 ( 1 ); impedance element Z H (k) has terminals Z H1 (k) and Z H2 (k); impedance element Z H ( 2 nMED ⁇ 1 ) has terminals Z H1 ( 2 nMED ⁇ 1 ) and Z H2 ( 2 nMED ⁇ 1 ); switch S H ( 1 ) has current conducting terminals S H1 ( 1 ) and S H2 ( 1 ) and a control terminal S H3 ( 1 ), switch S H (k) has terminals S H1 (k) and S H2 (k) and a control terminal S H3 (k); and switch S H ( 2 nMED ⁇ 1 ) has current conducting terminals S H1 ( 2 nMED ⁇ 1 ) and S H2 ( 2 nMED ⁇ 1 ) and a control terminal S H3 ( 2 nMED ⁇ 1
- Terminal Z HD1 of dummy impedance structure Z HD is connected to upper reference voltage terminal 32 and terminal Z HD2 is connected to terminal Z H1 ( 1 ) of impedance element Z H ( 1 ).
- Terminal Z H2 ( 1 ) is connected to terminal Z H1 (k)
- terminal Z H2 (k) is connected to terminal Z H1 ( 2 nMED ⁇ 1 )
- terminal Z H2 ( 2 nMED ⁇ 1 ) is the lower end of the impedance string connected to node 72 .
- S LH ( 2 nMED ⁇ 1 ), respectively, are commonly connected to reference terminal 32 .
- Current conducting terminal S H2 ( 1 ) of switch S H ( 1 ) is connected to terminal Z H2 ( 1 )
- current conducting terminal S H2 (k) is connected to terminal Z H2 (k)
- current conducting terminal S H2 ( 2 nMED ⁇ 1 ) is connected to node 72 .
- the number of impedance elements Z H of impedance stage 62 can range from 1 to 2 nMED ⁇ 1. Because impedance elements Z H ( 1 ) and Z H ( 2 nMED ⁇ 1 ) are shown in FIG. 2B (and FIG. 5B ), the index or variable “k” for impedance structure 62 can range from 2 to 2 nMED ⁇ 2.
- higher voltage medium range impedance stage 62 may be comprised of a dummy impedance structure Z HD and a single impedance element Z H ( 1 ).
- Dummy impedance structure Z HD is a circuit element connected in series with the chain or string of impedance elements Z H ( 1 ) to Z H ( 2 nMED ⁇ 1 ) that provides a constant equivalent impedance between the current conducting terminals S H1 (m) and S H2 (m) of any closed switch, i.e., a switch that is on, if all the other switches S H (j) under it are also closed or shorted. It should be noted that “j” is greater than or equal to 1 and less than or equal to “m.”
- the dummy structure is a resistor connected in parallel with a switch that is permanently on or closed.
- the resistance value of the dummy impedance depends on the value of the equivalent resistance of medium range impedance stage 62 and on the switch series on-resistance, R SH .
- the value of the dummy structure impedance is independent of the number of switches and whether they are shorted or not.
- m is the number of shorted switches in medium range impedance stage 62 and may range from zero to 2 nMED ;
- R MED is a medium resistor value
- R SHORT is a constant resistor representing the equivalent impedance between the current conducting terminals S H1 (m) and S H2 (m) of switch S H (m) for any “m” in the range 1 to 2 nMED ⁇ 1 and where R SHORT is in series with resistor R MED .
- resistor R SHORT ( R MED +( R MED *( R MED +4* R SH )) 1/2 )/2 EQT. 11
- R SH is the switch series on-resistance.
- a switch S H ( 1 ) having current conducting terminals S H1 ( 1 ) and S H2 ( 1 ) and a control terminal S H3 ( 1 ) is connected to impedance element Z H ( 1 ). More particularly, terminal S H1 ( 1 ) is connected to terminal Z HD ( 1 ) of dummy impedance element Z HD , terminal S H2 ( 1 ) is connected to terminal Z H2 ( 1 ) of impedance element Z H ( 1 ), and control terminal S H3 ( 1 ) of switch S H ( 1 ) is connected to terminal D H ( 1 ) of decoder 26 . Generally, a switch S H (k) having current conducting terminals S H1 (k) and S H2 (k) is connected to impedance element Z H (k).
- terminal S H1 (k) is connected to higher reference voltage terminal 32 and terminal S H2 (k) is connected to terminal Z H2 (k) of impedance element Z H (k).
- terminal S H1 ( 2 nMED ⁇ 1 ) is connected to higher reference voltage terminal 32 and terminal S H2 ( 2 nMED ⁇ 1 ) is connected to terminal Z H2 ( 2 nMED ⁇ 1 ) of impedance element Z H ( 2 nMED ⁇ 1 ).
- Control terminal S H3 ( 2 nMED ⁇ 1 ) is connected to terminal D H ( 2 nMED ⁇ 1 ) of decoder 26 .
- Connecting current conducting terminal S H2 ( 2 nMED ⁇ 1 ) to terminal Z H2 ( 2 nMED ⁇ 1 ) of impedance element Z H ( 2 nMED ⁇ 1 ) forms a node 72 .
- higher voltage LSB stage 60 may be comprised of a single impedance element and two switches such as, for example, impedance element Z WH ( 1 ) and switches S WH ( 1 ) and S WH ( 2 ).
- switch 46 has a current conducting terminal connected to node 71 and switch 48 has a current conducting terminal connected to node 73 .
- Nodes 73 and 71 are also referred to as upper and lower wiper terminals, respectively.
- the other current conducting terminals of switches 46 and 48 are commonly connected together to form a common wiper terminal 34 .
- the control terminal of switch 46 is connected to terminal D MSBL of decoder 26 and the control terminal of switch 48 is connected to terminal D MSBH of decoder 26 .
- FIGS. 3A , 3 B, 3 C, 3 D, 3 E, and 3 F illustrate a variable impedance network 14 A configured to have 256 taps and which receives a 32-bit decoded signal from an 8-bit decoder 26 .
- Reference character “A” has been appended to reference character “14” to distinguish a digital programmable potentiometer having 256 taps from one having more or fewer than 256 taps.
- Variable impedance network 14 A includes lower voltage impedance stage 40 , MSB stage 42 , and higher voltage impedance stage 44 .
- lower voltage impedance stage 40 is comprised of lower voltage LSB stage 50 and lower voltage medium range stage 52
- impedance stage 44 is comprised of higher voltage LSB stage 60 and higher voltage medium range stage 62 .
- Lower voltage medium range stage 52 includes a string of impedance elements Z L ( 1 ), Z L ( 2 ), Z L ( 3 ), Z L ( 4 ), Z L ( 5 ), Z L ( 6 ), and Z L ( 7 ) that are connected in a series configuration, i.e., a terminal of impedance element Z L ( 1 ) is connected to lower reference terminal 30 through a dummy structure Z LD and an opposing terminal of impedance element Z L ( 1 ) is connected to a terminal of impedance element Z L ( 2 ) to form a node 102 .
- An opposing terminal of impedance element Z L ( 2 ) is connected to a terminal of impedance element Z L ( 3 ) to form a node 104 and an opposing terminal of impedance element Z L ( 3 ) is connected to a terminal of impedance element Z L ( 4 ) to form a node 106 .
- An opposing terminal of impedance element Z L ( 4 ) is connected to a terminal of impedance element Z L ( 5 ) to form a node 108 and an opposing terminal of impedance element Z L ( 5 ) is connected to a terminal of impedance element Z L ( 6 ) to form a node 110 .
- An opposing terminal of impedance element Z L ( 6 ) is connected to a terminal of impedance element Z L ( 7 ) to form a node 112 and an opposing terminal of impedance element Z L ( 7 ) is connected to a terminal of impedance element Z WL ( 1 ) to form a node 114 .
- Lower voltage medium stage 52 further includes switches S L ( 1 ) to S L ( 7 ) connected to the string of impedance elements Z L ( 1 ) to Z L ( 7 ). More particularly, switch S L ( 1 ) has a terminal connected to node 102 , a terminal connected to terminal 30 , and a control terminal coupled for receiving a control signal via terminal D L ( 1 ); switch S L ( 2 ) has a terminal connected to node 104 , a terminal connected to terminal 30 , and a control terminal coupled for receiving a control signal via terminal D L ( 2 ); switch S L ( 3 ) has a terminal connected to node 106 , a terminal connected to terminal 30 , and a control terminal coupled for receiving a control signal via terminal D L ( 3 ); switch S L ( 4 ) has a terminal connected to node 108 , a terminal connected to terminal 30 , and a control terminal coupled for receiving a control signal via terminal D L ( 4 ); switch S L ( 5 ) has a terminal connected to node 110
- Lower voltage LSB stage 50 includes a string of impedance elements Z WL ( 1 ) to Z WL ( 15 ) that are coupled in a series configuration, i.e., a terminal of impedance element Z WL ( 1 ) is connected to node 114 and an opposing terminal of impedance element Z WL ( 1 ) is connected to a terminal of impedance element Z WL ( 2 ) to form a node 116 .
- An opposing terminal of impedance element Z WL ( 2 ) is connected to a terminal of impedance element Z WL ( 3 ) to form a node 118 and an opposing terminal of impedance element Z WL ( 3 ) is connected to a terminal of impedance element Z WL ( 4 ) to form a node 120 .
- An opposing terminal of impedance element Z WL ( 4 ) is connected to a terminal of impedance element Z WL ( 5 ) to form a node 122 and an opposing terminal of impedance element Z WL ( 5 ) is connected to a terminal of impedance element Z WL ( 6 ) to form a node 124 .
- An opposing terminal of impedance element Z WL ( 6 ) is connected to a terminal of impedance element Z WL ( 7 ) to form a node 126 and an opposing terminal of impedance element Z WL ( 7 ) is connected to a terminal of impedance element Z WL ( 8 ) to form a node 128 .
- An opposing terminal of impedance element Z WL ( 8 ) is connected to a terminal of impedance element Z WL ( 9 ) to form a node 130 and an opposing terminal of impedance element Z WL ( 9 ) is connected to a terminal of impedance element Z WL ( 10 ) to form a node 132 .
- An opposing terminal of impedance element Z WL ( 10 ) is connected to a terminal of impedance element Z WL ( 11 ) to form a node 134 and an opposing terminal of impedance element Z WL ( 11 ) is connected to a terminal of impedance element Z WL ( 12 ) to form a node 136 .
- An opposing terminal of impedance element Z WL ( 12 ) is connected to a terminal of impedance element Z WL ( 13 ) to form a node 138 and an opposing terminal of impedance element Z WL ( 13 ) is connected to a terminal of impedance element Z WL ( 14 ) to form a node 140 .
- An opposing terminal of impedance element Z WL ( 14 ) is connected to a terminal of impedance element Z WL ( 15 ) to form a node 142 .
- An opposing terminal of impedance element Z WL ( 15 ) is connected to terminal 42 A of MSB stage 42 .
- Terminal 42 B of MSB stage 42 is connected to an impedance element Z H ( 1 ) (shown in FIG. 3D ).
- Lower voltage LSB stage 50 further includes switches S WL ( 1 ) to S WL ( 17 ) connected to the string of impedance elements Z WL ( 1 ) to Z WL ( 16 ). More particularly, switch S WL ( 1 ) has a terminal connected to node 114 , a terminal connected to switch S MSBL to form a node 148 , and a control terminal coupled for receiving a control signal via terminal D WL ( 1 ); switch S WL ( 2 ) has a terminal connected to node 116 , a terminal connected to node 148 , and a control terminal coupled for receiving a control signal via terminal D WL ( 2 ); switch S WL ( 3 ) has a terminal connected to node 118 , a terminal connected to node 148 , and a control terminal coupled for receiving a control signal via terminal D WL ( 3 ); switch S WL ( 4 ) has a terminal connected to node 120 , a terminal connected to node 148 , and a control terminal coupled for receiving
- Higher voltage LSB stage 60 includes a string of impedance elements Z WH ( 1 ) to Z WH ( 16 ) that are coupled in a series configuration, i.e., a terminal of impedance element Z WH ( 1 ) is connected to node 150 and an opposing terminal of impedance element Z WH ( 1 ) is connected to a terminal of impedance element Z WH ( 2 ) to form a node 152 .
- An opposing terminal of impedance element Z WH ( 2 ) is connected to a terminal of impedance element Z WH ( 3 ) to form a node 154 and an opposing terminal of impedance element Z WH ( 3 ) is connected to a terminal of impedance element Z WH ( 4 ) to form a node 156 .
- An opposing terminal of impedance element Z WH ( 4 ) is connected to a terminal of impedance element Z WH ( 5 ) to form a node 158 and an opposing terminal of impedance element Z WH ( 5 ) is connected to a terminal of impedance element Z WH ( 6 ) to form a node 160 .
- An opposing terminal of impedance element Z WH ( 6 ) is connected to a terminal of impedance element Z WH ( 7 ) to form a node 162 and an opposing terminal of impedance element Z WH ( 7 ) is connected to a terminal of impedance element Z WH ( 8 ) to form a node 164 .
- An opposing terminal of impedance element Z WH ( 8 ) is connected to a terminal of impedance element Z WH ( 9 ) to form a node 166 and an opposing terminal of impedance element Z WH ( 9 ) is connected to a terminal of impedance element Z WH ( 10 ) to form a node 168 .
- An opposing terminal of impedance element Z WH ( 10 ) is connected to a terminal of impedance element Z WH ( 11 ) to form a node 170 and an opposing terminal of impedance element Z WH ( 11 ) is connected to a terminal of impedance element Z WH ( 12 ) to form a node 172 .
- An opposing terminal of impedance element Z WH ( 12 ) is connected to a terminal of impedance element Z WH ( 13 ) to form a node 174 and an opposing terminal of impedance element Z WH ( 13 ) is connected to a terminal of impedance element Z WH ( 14 ) to form a node 176 .
- An opposing terminal of impedance element Z WH ( 14 ) is connected to a terminal of impedance element Z WH ( 15 ) to form a node 178 .
- An opposing terminal of impedance element Z WH ( 15 ) (shown in FIG. 3D ) is connected to a terminal of impedance element Z H ( 7 ) to form a node 180 .
- Higher voltage LSB stage 60 further includes switches S WH ( 1 ) to S WH ( 16 ) connected to the string of impedance elements Z WH ( 1 ) to Z WH ( 16 ). More particularly, switch S WH ( 1 ) has a terminal connected to node 150 , an opposing terminal connected to a terminal of a switch S MSBH to form a node 184 , and a control terminal coupled for receiving a control signal via terminal D WH ( 1 ); switch S WH ( 2 ) has a terminal connected to node 152 , a terminal connected to node 184 , and a control terminal coupled for receiving a control signal via terminal D WH ( 2 ); switch S H ( 3 ) has a terminal connected to node 154 , a terminal connected to node 184 , and a control terminal coupled for receiving a control signal via terminal D WH ( 3 ); switch S WH ( 4 ) has a terminal connected to node 156 , a terminal connected to node 184 , and
- Upper voltage medium range stage 62 includes a string of impedance elements Z H ( 1 ) to Z H ( 7 ) that are coupled in a series configuration, i.e., a terminal of impedance element Z H ( 1 ) is coupled to upper reference terminal 32 through a dummy structure Z H (D) and an opposing terminal of impedance element Z H ( 1 ) is connected to a terminal of impedance element Z H ( 2 ) to form a node 196 .
- An opposing terminal of impedance element Z H ( 2 ) is connected to a terminal of impedance element Z H ( 3 ) to form a node 194 and an opposing terminal of impedance element Z H ( 3 ) is connected to a terminal of impedance element Z H ( 4 ) to form a node 192 .
- An opposing terminal of impedance element Z H ( 4 ) is connected to a terminal of impedance element Z H ( 5 ) to form a node 190 and an opposing terminal of impedance element Z H ( 5 ) is connected to a terminal of impedance element Z H ( 6 ) to form a node 188 .
- An opposing terminal of impedance element Z H ( 6 ) is connected to a terminal of impedance element Z H ( 7 ) to form a node 186 and an opposing terminal of impedance element Z H ( 7 ) is connected to node 182 .
- Higher voltage medium stage 62 further includes switches S H ( 1 ) to S H ( 7 ) connected to the string of impedance elements Z H ( 1 ) to Z H ( 7 ). More particularly, switch S H ( 1 ) has a terminal connected to node 196 , a terminal connected to terminal 32 , and a control terminal coupled for receiving a control signal via terminal D H ( 1 ); switch S H ( 2 ) has a terminal connected to node 194 , a terminal connected to terminal 32 , and a control terminal coupled for receiving a control signal via terminal D H ( 2 ); switch S H ( 3 ) has a terminal connected to node 192 , a terminal connected to Higher voltage medium stage 62 further includes switches S H ( 1 ) to S H ( 7 ) connected to the string of impedance elements Z H ( 1 ) to Z H ( 7 ).
- switch S H ( 4 ) has a terminal connected to node 190 , a terminal connected to terminal 32 , and a control terminal coupled for receiving a control signal via terminal D H ( 4 );
- switch S H ( 5 ) has a terminal connected to node 188 , a terminal connected to terminal 32 , and a control terminal coupled for receiving a control signal via terminal D H ( 5 );
- switch S H ( 6 ) has a terminal connected to node 186 , a terminal connected to terminal 32 , and a control terminal coupled for receiving a control signal via terminal D H ( 6 );
- switch S H ( 7 ) has a terminal connected to node 182 , a terminal connected to terminal 32 , and a control terminal coupled for receiving a control signal via terminal D H ( 7 ).
- control/decoder block 12 receives an input signal at input terminal 18 and extracts an n-bit signal or wiper address that is indicative of a desired impedance of variable impedance network 14 A and a desired voltage across variable impedance network 14 A.
- control/decoder block 12 extracts an 8-bit wiper address.
- Decoder 26 parses the 8-bit wiper address into segments that control stages 40 and 44 and switches S MSBL and S MSBH of variable impedance network 14 A.
- an 8-bit wiper address is divided such that the most significant bit controls switches S MSBL and S MBSH , the four least significant bits control the switches of the high resolution substages 40 and 60 , and the three bit section between the most significant bit and the four least significant bits control switches of the medium range substages 52 and 62 .
- an 8-bit wiper address controls the configuration of the switches of stages 40 and 44 and switches S MSBL and S MSBH .
- High resolution stages 40 and 60 are controlled by the 4-bit LSB section (WA[3]:WA[0]) of the 8-bit wiper address.
- stages 40 and 60 are substantially identical and are symmetrically positioned on each side of the bulk impedance Z B .
- Stages 40 and 60 each comprise sixteen tapping switches and fifteen impedance elements.
- the medium resolution stages 52 and 62 are controlled by the middle-range 3-bit section (WA[6]:WA[4]) of the 8-bit wiper address.
- stages 52 and 62 are substantially identical and are symmetrically positioned on each side of the bulk impedance Z B .
- Stages 52 and 62 each comprise seven shunt switches and seven resistors.
- stages 52 and 62 include a dummy structure such as, for example, a resistor in parallel with a switch that is permanently on.
- Stage 42 is a low resolution stage comprising a bulk impedance Z B having a relative position with respect to the wiper and that is controlled by one bit, i.e., the most significant bit WA[7], of the 8-bit wiper address and switches S MSBL and S MSBH .
- switch S MSBL when the wiper position is located on the lower half of variable impedance network 14 A, switch S MSBL is on or closed and switch S MSBH is off or open and when the wiper position is located on the upper half of variable impedance network 14 A, switch S MSBL is off or open and switch S MSBH is on or closed.
- FIGS. 4A and 4B illustrate the switch configurations for the switches S L ( 1 ) to S L ( 7 ), S WL ( 1 ) to S WL ( 16 ), S H ( 1 ) to S H ( 7 ), S WH ( 1 ) to S WH ( 16 ), S MSBL , and S MSBH for various values of the variable k that represent the corresponding tap position of the common wiper terminal between taps 0 and 255. It should be noted that switch configurations indicated by the number 1 are closed and switch configurations indicated by the number 0 are open.
- variable impedance network 14 B a circuit schematic of variable impedance network 14 B is shown. It should be understood that FIG. 5A shows a portion of variable impedance network 14 B and FIG. 5B shows another portion of variable impedance network 14 B. Accordingly, FIGS. 5A and 5B together show an embodiment of variable impedance network 14 and are therefore described together.
- Variable impedance network 14 B is similar to variable impedance network 14 except that dummy impedance structures Z LD and Z HD are not included in variable impedance network 14 B.
- terminal Z L1 ( 1 ) of impedance element Z L ( 1 ) is connected to or serves as terminal 30 and terminal Z H1 ( 1 ) of impedance element Z H ( 1 ) is connected to or serves as terminal 32 .
- the operation of variable impedance network 14 B is similar to that of variable impedance network 14 .
- the programmable multistage digital potentiometer is comprised of a plurality of stages connected as a string of stages and including a low resolution stage, two high resolution stages, and two medium resolution stages.
- the programmable multistage digital potentiometer receives an input signal and extracts an n-bit wiper address.
- a decoder circuit within the digital potentiometer assigns the bits to control the corresponding stages.
- the low resolution stages relative position with respect to the common wiper position is controlled by the most significant bit, divides the potentiometer into two portions, and is preferably positioned between a lower stage comprising a high resolution LSB stage and a medium resolution stage and a higher stage comprising another high resolution LSB stage and another medium resolution stage. Selecting the lowest value, i.e., 1-bit, for the lowest resolution stage minimizes the number of switches that are implemented, which lowers the area consumed by the switches and the cost of the potentiometer.
- the low resolution MSB stage is composed of a single bulk resistor having a value equal to (2 nMED ⁇ 1)*2 nLSB +1)*R 0 , where R 0 is a resistance value of an impedance element.
- each high resolution LSB stage has a terminal connected to one terminal of the MSB stage, a terminal connected to the medium resolution stage, and a terminal connected to the wiper terminal.
- the high resolution LSB stages are comprised of a chain of 2 nLSB ⁇ 1 impedance elements and 2 nLSB switches connected between a high resolution wiper terminal and every node of the chain of impedance elements.
- a wiper switch is connected between the common wiper terminal and a high resolution wiper terminal.
- each medium resolution stage is coupled between a corresponding variable impedance stage terminal and a high resolution LSB stage.
- each medium resolution stage is comprised of a chain of 2 nMED ⁇ 1 impedance elements where 2 nMED ⁇ 2 of them have substantially the same value and 2 nMED ⁇ 1 switches, where one current conducting terminal of each switch is coupled to a corresponding node of the chain of 2 nMED ⁇ 1 impedance elements and the other current conducting terminals are commonly connected together.
- each medium resolution stage further includes a dummy structure coupled to one of the impedance elements.
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Abstract
Description
n=n MSB +n MED +n LSB EQT. 1
nLSB=nMED; or EQT. 2
n LSB =n MED+1. EQT. 3
n=1+[(n−1)/2]+[n/2] EQT. 4
nMSB=1 EQT. 5
n MED=[(n−1)/2] EQT. 6
n LSB =[n/2] EQT. 7
R EQV(m)=(2nMED−1−m)*R MED +R SHORT EQT. 8
R SHORT=(R MED+(R MED*(R MED+4*R SH))1/2)/2 EQT. 9
R EQV(m)=(2nMED−1−m)*R MED +R SHORT EQT. 10
R SHORT=(R MED+(R MED*(R MED+4*R SH))1/2)/2 EQT. 11
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